Can we always find $X$ such that $X[FF^T]_+=[FF^T]_-$ for arbitrary matrix $F$?

Question

I know that it is not always possible to find $X$ that satisfies $XB=C$ (e.g. take $B=0$ and $C\neq0$)

But what if $B=[FF^T]_+$ and $C=[FF^T]_-$ for arbitrary $F$? where $[A]_+$ ($[A]_-$) is the positive (negative) part of the components of $A$

I am assuming all numbers to be real.

So the question is: for arbitrary real $F$ (square or rectangular) can we always find $X$ such that $X[FF^T]_+=[FF^T]_-$?

If so, how to prove it? If not, can you find a counter-example?

My intuition tells me that this is not always true but all examples I tried turned out to be true.